distinct lineages remaining in the source deme once we account for this initial set of coalescences. These lineages will undergo a random sequence of migration and coalescence events until there
is only one lineage remaining within the source deme. For example, if n
1
= 4, then one possible outcome would see one lineage migrate out of the deme followed by a pair of binary mergers, leaving
only one lineage within the source deme. Whatever the sequence, the amount of time required to scatter the lineages into different demes will be of order O1, whereas the time until either
the next mass extinction event or the next binary merger involving lineages outside of the source deme will be of order OD. Thus, if we again rescale time by a factor of D, then any sequence of
coalescence and migration events involving a source deme will effectively be instantaneous when we let D tend to infinity. This is the second way in which multiple merger events can arise in this
model. Furthermore, varying the migration rate and deme size changes not only the overall rate of coalescence, but also the relative rates of the different kinds of multiple merger events that can
occur. For example, if N m is very small, then the coalescent process will be close to a Λ-coalescent which has multiple mergers, but not simultaneous multiple mergers because most lineages that
are collected into a source deme by a mass extinction event will coalesce before any escape by migration. However, as N increases, so will the probability that multiple lineages enter into and
then escape from the source deme without coalescing. This suggests that at moderate values of N m, mass extinctions may be likely to result in simultaneous mergers i.e., the coalescent is a Ξ-
coalescent, while for very large values of N m, multiple mergers of all types will be unlikely and the coalescent process will tend towards Kingman’s coalescent.
1.2 Neutral genealogies and coalescents
In the last twenty years, coalescent processes have taken on increasingly important role in both theoretical and applied population genetics, where their relationship to genealogical trees has made
them powerful tools to study the evolution of genetic diversity within a population. Under the assumption of neutrality, allelic types do not influence the reproduction of individuals and it is
therefore possible to separate ‘type’ and ‘descent’. This allows us to study the genealogy of a sample of individuals on its own and then superimpose a mechanism describing how types are transmitted
from parent to offspring, justifying the interest in investigating genealogical processes corresponding to particular reproduction mechanisms without explicit mention of types. We refer to Nordborg
[2001] for a review of coalescent theory in population genetics.
Beginning with the coalescent process introduced by Kingman [1982] to model the genealogy of a sample of individuals from a large population, three increasingly general classes of coalescent
processes have been described. A key feature shared by all three classes is the following consistency property: the process induced on the set of all partitions of {1, . . . , n} by the coalescent acting on the
partitions of {1, . . . , n+k} obtained by considering only the blocks containing elements of {1, . . . , n} has the same law as the coalescent acting on the partitions of {1, . . . , n}. In terms of genealogies,
this property means that the genealogy of n individuals does not depend on the size of the sample that contains them. To describe these continuous-time Markov processes, it will be convenient to
introduce some notation. For all n ∈ N, we denote the set of all partitions of [n] ≡ {1, . . . , n} by P
n
. In the following, the index n of the set of partitions in which we are working will be referred to as the
sample size, an element of {1, . . . , n} will be called an individual, and ‘block’ or ‘lineage’
will be equivalent terminology to refer to an equivalence class. If ζ ∈
S
n
P
n
, then | ζ| = k means
that the partition ζ has k blocks. Also, for ζ, η ∈ P
n
and k
1
, . . . , k
r
≥ 2, we will write η ⊂
k
1
,...,k
r
ζ if 247
η is obtained from ζ by merging exactly k
1
blocks of ζ into one block, k
2
into another block, and so on. Kingman’s coalescent is defined on P
n
for all n ≥ 1, as a Markov process with the following Q-matrix: if
ζ, η ∈ P
n
, q
K
ζ → η =
1 if
η ⊂
2
ζ, −
|ζ| 2
if η = ζ,
otherwise. A more general class of exchangeable coalescents, allowing mergers of more than two blocks at a
time, was studied by Pitman [1999] and Sagitov [1999]. These coalescents with multiple mergers or Λ-coalescents are in one-to-one correspondence with the finite measures on [0, 1] in the fol-
lowing manner: for a given coalescent, there exists a unique finite measure Λ on [0, 1] such that the entries q
Λ
ζ → η of the Q-matrix of the coalescent, for ζ, η ∈ P
n
, are given by
q
Λ
ζ → η =
R
1
Λd xx
k−2
1 − x
b−k
if η ⊂
k
ζ and |ζ| = b, −
R
1
Λd xx
−2
1 − 1 − x
b−1
1 − x + b x if
η = ζ and |ζ| = b, otherwise.
Kingman’s coalescent is recovered by taking Λ = δ
, the point mass at 0. Lastly, a third and wider class of coalescents was introduced by Möhle and Sagitov [2001] and Schweinsberg [2000], for
which mergers involving more than one ancestor are allowed. These coalescents with simultaneous multiple mergers or Ξ-coalescents are characterized in Schweinsberg [2000] by a finite Borel
measure on the infinite ordered simplex
∆ = n
x
1
, x
2
, . . . : x
1
≥ x
2
≥ . . . ≥ 0,
∞
X
i=1
x
i
≤ 1 o
. Indeed, to each coalescent corresponds a unique finite measure Ξ on ∆ of the form Ξ = Ξ
+ aδ ,
where Ξ has no atom at zero and a ∈ [0, ∞, such that the transition rates of the coalescent acting
on P
n
are given by q
Ξ
ζ → η = Z
∆
Ξ dx
P
∞ j=1
x
2 j
s
X
l=0
X
i
1
6=...6=i
r+l
s l
x
k
1
i
1
. . . x
k
r
i
r
x
i
r+1
. . . x
i
r+l
1 −
∞
X
j=1
x
j s−l
+ a I
{r=1,k
1
=2}
if η ⊂
k
1
,...,k
r
ζ and s ≡ |ζ| − P
r i=1
k
i
. The other rates for η 6= ζ are equal to zero. The Λ-coalescents
are particular cases of Ξ-coalescents, for which Ξx
2
0 = 0. As mentioned above, coalescent processes can be used to describe the genealogy of large popu-
lations. Indeed, a large body of literature has been devoted to describing conditions on the de- mography of a population of finite size N that guarantee that the genealogical process of a sample
of individuals converges to a coalescent as N tends to infinity. Such limiting results for populations with discrete non-overlapping generations are reviewed in Möhle [2000], and some examples can be
found for instance in Schweinsberg [2003], Eldon and Wakeley [2006] and Sargsyan and Wakeley [2008]. In these examples, the shape of the limiting coalescent is related to the propensity of indi-
viduals to produce a non-negligible fraction of the population in the next generation.
However, the representation of the genealogy as a coalescent requires in particular that any pair of lineages has the same chance to coalesce. This condition breaks down when the population is
248
structured into subpopulations, since then coalescence will occur disproportionately often between lineages belonging to the same deme. To model these kinds of scenarios, structured analogues of
coalescent processes were introduced [see e.g. Notohara, 1990; Wilkinson-Herbots, 1998], which allow lineages both to move between demes as well as coalesce within demes. Various state spaces
have been used to describe a structured coalescent, such as vectors in which the i’th component gives the lineages or their number present in deme i, or vectors of pairs ‘block × deme label’.
All these representations of a structured genealogy take into account the fact that the reproductive or dispersal dynamics may differ between demes, hence the need to keep track of the location of
the lineages. In contrast, several papers investigate models where the structure of the genealogy collapses on an appropriate time scale, i.e., the limiting genealogy no longer sees the geographical
division of the population. In Cox [1989], demes are located at the sites of the torus TD ⊂ Z
d
of size D and each site can contain at most one lineage. Lineages move between sites according to a simple random walk, and when one of them lands on a site already occupied, it merges in-
stantaneously with the inhabitant of this ‘deme’. These coalescing random walks, dual to the voter model on the torus, are proved to converge to Kingman’s coalescent as D → ∞. More precisely,
Cox shows that if n
∞ lineages start from n sites independently and uniformly distributed over T
D, then the process counting the number of distinct lineages converges to the pure death pro- cess that describes the number of lineages in Kingman’s coalescent. This analysis is generalized in
Cox and Durrett [2002] and Zähle et al. [2005], where each site of the torus now contains N ∈ N individuals and a Moran-type reproduction dynamics occurs within each deme. Again, the limiting
genealogy of a finite number of particles sampled at distant sites is given by Kingman’s coalescent, and convergence is in the same sense as for Cox’ result. Other studies of systems of particles mov-
ing between discrete subpopulations and coalescing do not require that the initial locations of the lineages be thinned out. In Greven et al. [2007], demes are distributed over the grid Z
2
and the process starts with a Poisson-distributed number of lineages on each site of a large box of size D
α2
, for some
α ∈ 0, 1]. The authors show that the total number of lineages alive at times of the form D
t
converges in distribution as a process indexed by t ≥ α to a time-change of the block counting
process of Kingman’s coalescent, as D → ∞. See Greven et al. [2007] for many other references related to these ideas.
Our emphasis in this paper will be on the separation of time scales phenomenon and the way in which local and global demographic processes jointly determine the statistics of the limiting
coalescent process. Consequently, we shall always assume that the demes comprising our population are exchangeable, i.e., the same demographic processes operate within each deme, and migrants are
equally likely to come from any one of the D demes. In this simplified setting, we only need to know how lineages are grouped into demes, but not the labels of these demes.
1.3 Separation of time scales
